Question

A cart on frictionless rollers approaches a smooth, curved slope h = 0.8 meters high. What minimum speed v is required for the cart to reach the top of the slope?___ m/s

Answer 1

We are given that a frictionless car must go up a height "h". To determine the minimum velocity we need to use a balance of energy between the point where the car starts to go up the slope and the point where it reached the height. At the first point, the type of energy in the car is kinetic energy and when it reaches the height it has potential energy, therefore:

$E_k=E_p$

Kinetic energy is given by:

$E_k=\frac{1}{2}mv^2_{}$

Where:

$\begin{gathered} m=\text{ mass} \\ v=\text{ velocity} \end{gathered}$

The potential energy is given by:

$E_p=\text{mgh}$

Where:

$\begin{gathered} g=\text{ acceleration of gravity } \\ h=\text{ height} \end{gathered}$

Substituting in the balance of energy:

$\frac{1}{2}mv^2=\text{mgh}$

We cancel out the mass:

$\frac{1}{2}v^2=gh$

Now we solve for the velocity first by multiplying both sides by 2:

$v^2=2gh$

Now we take the square root to both sides:

$v=\sqrt[]{2gh}$

Now we substitute the given values:

$v=\sqrt[]{2(9.8\frac{m}{s^2})(0.8m)}$

Solving the operations we get:

$v=3.96\frac{m}{s}$

Therefore, the minimum speed must be 3.96 meters per second.